 On the Sphere and Cylinder

On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BC.^{[1]} It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.^{[2]}
Contents
The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. In his work, Archimedes showed that the surface area of a cylinder is equal to:
and that the volume of the same is:
 ^{[3]}
On the sphere, he showed that the surface area is four times the area of its great circle. In modern terms, this means that the surface area is equal to:
The result for the volume of the contained ball stated that it is twothirds the volume of a circumscribed cylinder, meaning that the volume is
Archimedes was particularly proud of this latter result, and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave  he had shown that both the volume and the surface area of the sphere were twothirds that of the cylinder. Later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation.^{[4]}
The argument Archimedes used to prove the formula for the volume of a ball was rather involved in its geometry, and many modern textbooks have a simplified version using the concept of a limit, which, of course, did not exist in Archimedes' time. Archimedes used an inscribed halfpolygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he then determined the volume.^{[5]}
It seems that this is not the original method Archimedes used to derive this result, but the best formal argument available to him in the Greek mathematical tradition. His original method likely involved a clever use of levers. A palimsest rediscovered in the 19th century contained many of Archimedes works, including The Method of Mechanical Theorems, in which he describes a method to determine volumes which involves balances, centers of mass and infinitesimal slices .
Notes
 ^ Dunham 1990, p. 99
 ^ Weisstein, Eric W., "Sphere" from MathWorld. Retrieved on 20080622
 ^ Dunham 1994, p. 227
 ^ "Archimedes: His Works", Britannica Online, Encyclopedia Britannica, http://www.britannica.com/EBchecked/topic/32808/Archimedes/21480/Hisworks#ref=ref383380&tocpanel=sectionId~toc214869%2CtocId~toc214869, retrieved 20080623
 ^ (Dunham 1994, p. 226)
References
 Dunham, William (1990), Journey Through Genius (1st ed.), John Wiley and Sons, ISBN 0471500305
 Dunham, William (1994), The Mathematical Universe (1st ed.), John Wiley and Sons, ISBN 0471536563
 S.H. Gould, The Method of Archimedes, The American Mathematical Monthly. Vol. 62, No. 7 (Aug.  Sep., 1955), pp. 473476
Categories: Ancient Greek mathematical works
 Euclidean geometry
 Works by Archimedes
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